zahedpour_2015

S. Zahedpour, J. K. Wahlstrand and H. M. Milchberg, “Measurement of the nonlinear refractive index of air constituents at mid-infrared wavelengths”, Opt. Lett. 40, 5794 (2015)

Source record · Catalogue · Review register

DOI: 10.1364/OL.40.005794

Source kind: measurement

Recorded technique: Single-shot supercontinuum spectral interferometry with a tunable pump: an OPA (1100-2600 nm; 1250, 1650, 2200, 2400 nm used) or the 800-nm Ti:sapphire (and 400 nm) pump, 500 Hz; supercontinuum probe and reference (500-700 nm, from 2 atm Xe; chirped to ~1.5 ps), crossing the pump at 2 degrees in a backfilled chamber (interaction length L ~ 2 mm); probe parallel or perpendicular to the pump. Absolute n2 WITHOUT density or intensity measurements: the electronic phase is referenced to the rotational phase of N2 and O2, whose size is fixed by Delta alpha (wahlstrand_2012, 800 nm); Ar is referenced to N2 (pp. 5794-5796).

Materials in structured entries: Ar, N2, O2

Dataset quantities: nonlinear_index

Contents: 10 datasets; 2 formulas.

Curation notes

Journal pages 5794-5797 = PDF pages 1-4. Identity checked against the first page. Air is not curated (the title’s ‘air constituents’ are N2, O2, Ar).

DEFINING EQUATIONS (Gaussian). Separation, Eq. (1), p. 5795: Delta phi_elec = 3(Delta phi_par + 2 Delta phi_perp)/5, Delta phi_rot = 2(Delta phi_par - 3 Delta phi_perp)/5. The probe index (p. 5795): ‘Delta n_p(x, t) = 2 n2 I(x, t) + int R(t - t’) I(x, t’) dt’’, ‘where I is the pump intensity; n2 is the electronic Kerr coefficient, which depends on the probe and pump wavelengths’: n2 is the SELF coefficient. Eq. (2), p. 5795: Delta phi_elec = 2 n2 I0 f(x, t) k_p L; Delta phi_rot = (2 pi N k_p L / n0) Delta alpha(lambda_p) (<cos^2 theta>_t - 1/3) = (2 pi N k_p L / n0) Delta alpha(lambda_p) [Delta alpha(lambda_e) I0 / (c hbar)] int g(t - t’) f(x, t’) dt’, g the rescaled rigid-rotor impulse response; the fit gives mu1 = Delta alpha(lambda_p) Delta alpha(lambda_e) N L I0 and then mu2 = n2 (Delta alpha(lambda_p) Delta alpha(lambda_e))^-1 (p. 5796). Kerr coefficient wanted (p. 5796): ‘n2(omega_e) = (12 pi^2 / n0^2 c) N gamma(3)(omega_e; omega_e, -omega_e, omega_e)’; measured: ‘n2(omega_e, omega_p) proportional to gamma(3)(omega_p; omega_e, -omega_e, omega_p)’. Dispersion, Eq. (3), p. 5796: gamma(3)(omega_s; omega1, omega2, omega3) = gamma0 (1 + a omega_L^2 + b omega_L^4 + ...), omega_L^2 = omega_s^2 + omega1^2 + omega2^2 + omega3^2; for the pump-probe case omega_L^2 = 2 omega_e^2 + 2 omega_p^2; the difference n2(omega_e, omega_p) against n2(omega_e) is < ~6 % for pump and probe longer than ~500 nm and at 400/600 nm (p. 5796).

DELTA ALPHA DISPERSION (p. 5796): Delta alpha(omega) ~ Delta alpha(0) + C omega^2 (Spackman 1991); Alms et al.’s light scattering for N2 is ‘well fit by Delta alpha(0) = 6.6 x 10^-25 cm3 and C = 3.8 x 10^-57 cm3 s2’, rescaled so that Delta alpha(800 nm) equals wahlstrand_2012’s value; Delta alpha(800)/Delta alpha(0) ~ 1.03, Delta alpha(400)/Delta alpha(0) ~ 1.13. Table 1 note b: N2 adjusted for this dispersion; note c: not adjusted (Delta alpha(800 nm) of [22] used at all wavelengths) - N2 both ways, O2 unadjusted; note d: Ar referenced to the N2 measurements (which N2 column is not stated). Neglecting the dispersion changes N2 by at most ~10 % near 400 nm and < 2 % elsewhere.

REFERENCE STATE: ‘Scaled to Atmospheric Pressure’ (Table 1 caption), no temperature: 101325 Pa, 293.15 K assumed. Note the absolute scale is that of wahlstrand_2012’s Delta alpha (itself on the ~291-K N2 flow calibration, TC-281).

UNCERTAINTY (p. 5796): the residual of the mu1 fits (standard deviation over x), laser power drift, and the propagated uncertainty of Delta alpha from [22]; larger at 2400 nm (phase shifts <= ~30 mrad). Level not stated.

PULSES. 40 fs at 800 nm (p. 5795); the OPA pulse durations are not stated (the 2200-nm envelope is ‘longer and more complex’, Fig. 3). Probe central wavelength 600 nm (p. 5795). Short pulses (< ~50 fs) see mainly the electronic response in N2 and O2, long ones (> ~150 fs) mainly the rotational one (p. 5796).

THE [32] COLUMN (Shelton and Rice; computed with Eq. (3) and the ESHG gamma0, a, b of [33] = shelton_1990). Check (2026-09-26): Shelton 1990 Table IV’s gamma0, A, B (library table, N2 57.20, 1.003, 1.852; O2 60.03, 1.112, 4.603; Ar 72.75, 1.066, 2.033) with a = A/(2 pi c)^2, b = B/(2 pi c)^4, omega_L^2 = 2 omega_e^2 + 2 omega_p^2 (probe 600 nm) reproduce all 18 printed values to +-0.5 % (the printed precision) with one common factor, n2 = 0.133 x gamma(10^-63 C^4 m^4 J^-3) x 10^-20 cm2/W; this equals N gamma / (8 eps0^2 c) with N the ideal-gas density at 101325 Pa and 293.15 K (0.1331). The self-Kerr omega_L^2 = 4 omega_e^2 does not reproduce the 400-nm values (-12 to -16 %). So the column is n2(omega_e, omega_p) of the T-convention gamma at 1 atm and ~293 K. use = ‘check’.

PRINTED COEFFICIENTS (formulas below): ‘a = 1.8 x 10^-33, 3 x 10^-33 s^2, b = 1.5 x 10^-65, 1.6 x 10^-65 s^4, and gamma0 = 7.7 x 10^-30, 9.8 x 10^-30 cm^6 W^-1 s^-1 for N2 and Ar’. From Shelton’s A, B: a(N2) = 2.83e-33 s^2 (printed 1.8e-33: misprint, 2.8 intended - the [32] column is only reproduced with 2.83e-33), a(Ar) = 3.00e-33, b(N2) = 1.47e-65, b(Ar) = 1.61e-65 s^4. The printed gamma0 are in the ratio of Shelton’s (0.786), but with the paper’s n2 = (12 pi^2 / n0^2 c) N gamma they give n2 ten times the [32] column (7.6e-19 against 7.6e-20 cm2/W for N2 at 1 atm, 293 K): the gamma0 values or their unit are off by a factor 10 (TC-281); not corrected here.

Recorded bibliographic identity status: VALIDATED. This concerns identification of the source; it does not establish the scientific correctness of its contents.

Datasets

zahedpour_2015/n2_adjusted_N2

{
  "id": "n2_adjusted_N2",
  "material": "N2",
  "file": "N2/zahedpour_2015__n2_adjusted_N2.csv",
  "table": "tbl_818914a52f6f",
  "quantity": "nonlinear_index",
  "unit": "cm2/W",
  "scale": 1e-20,
  "abscissa_unit": "nm",
  "state": {
    "pressure_pa": 101325.0,
    "temperature_k": 293.15
  },
  "use": "primary",
  "uncertainty": {
    "column": true,
    "level": "not stated"
  },
  "parameters": {
    "part": "electronic_vibrational",
    "coefficient": "self",
    "probe_nm": 600
  }
}

Table 1 (p. 5797), N2 note b, ‘This Work’: adjusted for the dispersion of Delta alpha (Alms et al. fit, rescaled to 800 nm). Rows are pump wavelengths (vacuum assumed). Electronic self coefficient; the adiabatic vibrational part is not separated (40-fs pump at 800 nm; OPA durations not stated).

zahedpour_2015/n2_N2

{
  "id": "n2_N2",
  "material": "N2",
  "file": "N2/zahedpour_2015__n2_N2.csv",
  "table": "tbl_818914a52f6f",
  "quantity": "nonlinear_index",
  "unit": "cm2/W",
  "scale": 1e-20,
  "abscissa_unit": "nm",
  "state": {
    "pressure_pa": 101325.0,
    "temperature_k": 293.15
  },
  "use": "check",
  "uncertainty": {
    "column": true,
    "level": "not stated"
  },
  "parameters": {
    "part": "electronic_vibrational",
    "coefficient": "self",
    "probe_nm": 600
  }
}

Table 1 (p. 5797), N2 note c, ‘This Work’: the same measurement without the Delta alpha dispersion adjustment (Delta alpha(800 nm) at all wavelengths); n2_adjusted_N2 is the paper’s preferred analysis, this one ‘check’.

zahedpour_2015/n2_O2

{
  "id": "n2_O2",
  "material": "O2",
  "file": "O2/zahedpour_2015__n2_O2.csv",
  "table": "tbl_818914a52f6f",
  "quantity": "nonlinear_index",
  "unit": "cm2/W",
  "scale": 1e-20,
  "abscissa_unit": "nm",
  "state": {
    "pressure_pa": 101325.0,
    "temperature_k": 293.15
  },
  "use": "primary",
  "uncertainty": {
    "column": true,
    "level": "not stated"
  },
  "parameters": {
    "part": "electronic_vibrational",
    "coefficient": "self",
    "probe_nm": 600
  }
}

Table 1 (p. 5797), O2 note c, ‘This Work’: not adjusted for Delta alpha dispersion (none available for O2). Rows are pump wavelengths.

zahedpour_2015/n2_Ar

{
  "id": "n2_Ar",
  "material": "Ar",
  "file": "Ar/zahedpour_2015__n2_Ar.csv",
  "table": "tbl_818914a52f6f",
  "quantity": "nonlinear_index",
  "unit": "cm2/W",
  "scale": 1e-20,
  "abscissa_unit": "nm",
  "state": {
    "pressure_pa": 101325.0,
    "temperature_k": 293.15
  },
  "use": "primary",
  "uncertainty": {
    "column": true,
    "level": "not stated"
  },
  "parameters": {
    "part": "electronic",
    "coefficient": "self",
    "probe_nm": 600
  }
}

Table 1 (p. 5797), Ar note d, ‘This Work’: referenced to the N2 measurements (which N2 column not stated).

zahedpour_2015/shelton_N2

{
  "id": "shelton_N2",
  "material": "N2",
  "file": "N2/zahedpour_2015__shelton_N2.csv",
  "table": "tbl_818914a52f6f",
  "quantity": "nonlinear_index",
  "unit": "cm2/W",
  "scale": 1e-20,
  "abscissa_unit": "nm",
  "state": {
    "pressure_pa": 101325.0,
    "temperature_k": 293.15
  },
  "use": "check",
  "reproduced_from": [
    "shelton_1990"
  ],
  "parameters": {
    "part": "unknown",
    "coefficient": "self",
    "probe_nm": 600
  }
}

Table 1 (p. 5797), ‘[32]’ column: n2 computed by the authors with Eq. (3) and Shelton 1990’s ESHG gamma0, A, B; reproduced here to +-0.5 % as n2(omega_e, omega_p) (probe 600 nm) = N gamma_T / (8 eps0^2 c) at 1 atm, 293.15 K (source notes). A calculation from shelton_1990’s coefficients: ‘check’. For N2 and O2 Shelton’s fit is to gamma^e (electronic) of Table III(a); for Ar the total.

zahedpour_2015/shelton_O2

{
  "id": "shelton_O2",
  "material": "O2",
  "file": "O2/zahedpour_2015__shelton_O2.csv",
  "table": "tbl_818914a52f6f",
  "quantity": "nonlinear_index",
  "unit": "cm2/W",
  "scale": 1e-20,
  "abscissa_unit": "nm",
  "state": {
    "pressure_pa": 101325.0,
    "temperature_k": 293.15
  },
  "use": "check",
  "reproduced_from": [
    "shelton_1990"
  ],
  "parameters": {
    "part": "unknown",
    "coefficient": "self",
    "probe_nm": 600
  }
}

Table 1 (p. 5797), ‘[32]’ column: n2 computed by the authors with Eq. (3) and Shelton 1990’s ESHG gamma0, A, B; reproduced here to +-0.5 % as n2(omega_e, omega_p) (probe 600 nm) = N gamma_T / (8 eps0^2 c) at 1 atm, 293.15 K (source notes). A calculation from shelton_1990’s coefficients: ‘check’. For N2 and O2 Shelton’s fit is to gamma^e (electronic) of Table III(a); for Ar the total.

zahedpour_2015/shelton_Ar

{
  "id": "shelton_Ar",
  "material": "Ar",
  "file": "Ar/zahedpour_2015__shelton_Ar.csv",
  "table": "tbl_818914a52f6f",
  "quantity": "nonlinear_index",
  "unit": "cm2/W",
  "scale": 1e-20,
  "abscissa_unit": "nm",
  "state": {
    "pressure_pa": 101325.0,
    "temperature_k": 293.15
  },
  "use": "check",
  "reproduced_from": [
    "shelton_1990"
  ],
  "parameters": {
    "part": "electronic",
    "coefficient": "self",
    "probe_nm": 600
  }
}

Table 1 (p. 5797), ‘[32]’ column: n2 computed by the authors with Eq. (3) and Shelton 1990’s ESHG gamma0, A, B; reproduced here to +-0.5 % as n2(omega_e, omega_p) (probe 600 nm) = N gamma_T / (8 eps0^2 c) at 1 atm, 293.15 K (source notes). A calculation from shelton_1990’s coefficients: ‘check’. For N2 and O2 Shelton’s fit is to gamma^e (electronic) of Table III(a); for Ar the total.

zahedpour_2015/n2_rot_adjusted_N2

{
  "id": "n2_rot_adjusted_N2",
  "material": "N2",
  "file": "N2/zahedpour_2015__n2_rot_adjusted_N2.csv",
  "table": "tbl_818914a52f6f",
  "quantity": "nonlinear_index",
  "unit": "cm2/W",
  "scale": 1e-20,
  "abscissa_unit": "nm",
  "state": {
    "pressure_pa": 101325.0,
    "temperature_k": 293.15
  },
  "use": "check",
  "parameters": {
    "part": "rotational",
    "coefficient": "self",
    "probe_nm": 600
  }
}

Table 1 (p. 5797), n2,rot (N2 note b group): the calculated long-pulse (adiabatic) rotational coefficient from the measured Delta alpha of [22] (wahlstrand_2012); n2,eff = n2 + n2,rot for very long pulses (note a). Derived: ‘check’.

zahedpour_2015/n2_rot_N2

{
  "id": "n2_rot_N2",
  "material": "N2",
  "file": "N2/zahedpour_2015__n2_rot_N2.csv",
  "table": "tbl_818914a52f6f",
  "quantity": "nonlinear_index",
  "unit": "cm2/W",
  "scale": 1e-20,
  "abscissa_unit": "nm",
  "state": {
    "pressure_pa": 101325.0,
    "temperature_k": 293.15
  },
  "use": "check",
  "parameters": {
    "part": "rotational",
    "coefficient": "self",
    "probe_nm": 600
  }
}

Table 1 (p. 5797), n2,rot (N2 note c group): the calculated long-pulse (adiabatic) rotational coefficient from the measured Delta alpha of [22] (wahlstrand_2012); n2,eff = n2 + n2,rot for very long pulses (note a). Derived: ‘check’.

zahedpour_2015/n2_rot_O2

{
  "id": "n2_rot_O2",
  "material": "O2",
  "file": "O2/zahedpour_2015__n2_rot_O2.csv",
  "table": "tbl_818914a52f6f",
  "quantity": "nonlinear_index",
  "unit": "cm2/W",
  "scale": 1e-20,
  "abscissa_unit": "nm",
  "state": {
    "pressure_pa": 101325.0,
    "temperature_k": 293.15
  },
  "use": "check",
  "parameters": {
    "part": "rotational",
    "coefficient": "self",
    "probe_nm": 600
  }
}

Table 1 (p. 5797), n2,rot (O2): the calculated long-pulse (adiabatic) rotational coefficient from the measured Delta alpha of [22] (wahlstrand_2012); n2,eff = n2 + n2,rot for very long pulses (note a). Derived: ‘check’.

Formulas

zahedpour_2015/bishop_dispersion_N2

{
  "id": "bishop_dispersion_N2",
  "material": "N2",
  "equation": "Eq. (3), p. 5796, with the coefficients in the text below it",
  "printed": "gamma(3)(omega_s; omega_1, omega_2, omega_3) = gamma_0 (1 + a omega_L^2 + b omega_L^4 + ...), omega_L^2 = omega_s^2 + omega_1^2 + omega_2^2 + omega_3^2; 'a = 1.8 x 10^-33 ... s^2, b = 1.5 x 10^-65 ... s^4, and gamma_0 = 7.7 x 10^-30 ... cm^6 W^-1 s^-1 for N2'",
  "variable": "omega: angular frequencies in s^-1; pump-probe case omega_L^2 = 2 omega_e^2 + 2 omega_p^2; self Kerr 4 omega^2",
  "state": "gamma per molecule; n2(omega_e) = (12 pi^2 / n0^2 c) N gamma (p. 5796)",
  "density": "none",
  "range": "below electronic resonance; used here 400-2400 nm pump, 600 nm probe",
  "coefficients": {
    "gamma0": "7.7e-30",
    "a": "1.8e-33",
    "b": "1.5e-65"
  },
  "check": "The coefficients are shelton_1990's ESHG gamma0, A, B rescaled to angular frequency (a = A/(2 pi c)^2, b = B/(2 pi c)^4): b reproduces (1.47e-65 from B = 1.852e-20 cm^4), a does not (2.83e-33 from A = 1.003e-10 cm^2; printed 1.8e-33). The paper's own [32] column is reproduced to +-0.5 % only with a = 2.83e-33 (checked 2026-09-26): the printed 1.8 is a misprint for 2.8. gamma0 is 10 times too large for the printed n2 formula (source notes)."
}

Reproduced and rescaled from shelton_1990 (Table IV, ESHG fit to gamma^e, He calibration of that paper); Bishop’s (1989) form. Not an independent result. Corrected form: a = 2.83e-33 s^2 (from Shelton’s A); gamma0 unresolved (factor 10).

zahedpour_2015/bishop_dispersion_Ar

{
  "id": "bishop_dispersion_Ar",
  "material": "Ar",
  "equation": "Eq. (3), p. 5796, with the coefficients in the text below it",
  "printed": "as bishop_dispersion_N2; 'a = ... 3 x 10^-33 s^2, b = ... 1.6 x 10^-65 s^4, and gamma_0 = ... 9.8 x 10^-30 cm^6 W^-1 s^-1 for ... Ar'",
  "variable": "omega: angular frequencies in s^-1; omega_L^2 as bishop_dispersion_N2",
  "state": "gamma per atom; n2(omega_e) = (12 pi^2 / n0^2 c) N gamma (p. 5796)",
  "density": "none",
  "range": "below electronic resonance; used here 400-2400 nm pump, 600 nm probe",
  "coefficients": {
    "gamma0": "9.8e-30",
    "a": "3e-33",
    "b": "1.6e-65"
  },
  "check": "From shelton_1990 Table IV (A = 1.066e-10 cm^2, B = 2.033e-20 cm^4): a = 3.00e-33 s^2, b = 1.61e-65 s^4, both reproduced. With these the [32] column is reproduced to +-0.5 % (checked 2026-09-26). gamma0 ratio Ar/N2 = 1.27 as Shelton's; absolute gamma0 a factor 10 too large for the printed n2 formula (source notes)."
}

Reproduced and rescaled from shelton_1990.

Corrections

No dataset errata are recorded in this source record.

Formula corrections, where recorded, appear with their formulas above. Curation notes may also discuss corrections or unresolved interpretation. Consult the review register for the checks and decisions.